1.12.15: Expansions- Isentropic- Solutions
The Gibbs energy of a closed system at thermodynamic equilibrium containing two chemical substances is defined by equation (a) where the molecular composition/organisation is signalled by \(\xi^{\mathrm{eq}}\).
\[\mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0, \xi^{\mathrm{eq}}\right] \nonumber \]
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0, \xi^{\mathrm{eq}}\right] \nonumber \]
\[\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0, \xi^{\mathrm{eq}}\right] \nonumber \]
A common feature is the use of the two intensive variables, temperature and pressure, in the definition of extensive properties \(\mathrm{G}\), \(\mathrm{V}\) and \(\mathrm{S}\). The properties \(\mathrm{G}\), \(\mathrm{V}\) and \(\mathrm{S}\) are Gibbsian.
The system is perturbed by an increase in temperature along a path such that the affinity for spontaneous change remains zero and the entropy remains equal to that defined by equation (c). In principle we plot volume \(\mathrm{V}\) as a function of temperature at constant \(\mathrm{n}_{1}\), \(\mathrm{n}_{2}\), at '\(\mathrm{A}=0\)' and at a constant entropy equal to that defined by equation (c). The gradient of the plot at the point where the volume is defined by equation (b) yields the equilibrium isentropic expansion, \(\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0)\) [1]; equation (d); isentropic = adiabatic and at equilibrium.
\[\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0)=(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{S} . \mathrm{A}=0} \nonumber \]
\(\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0)\) characterises the system defined by the Gibbsian set of independent variables \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0, \xi^{\mathrm{eq}}\right]\). As we change the amount of solute \(\mathrm{n}_{j}\) for a fixed temperature, pressure and amount of solvent \(\mathrm{n}_{1}\), so both \(\mathrm{V}(\mathrm{aq})\) and \(\mathrm{S}(\mathrm{aq})\) change yielding a new isentropic thermal expansion \(\mathrm{E}_{\mathrm{S}}(\mathrm{aq})\) at a new entropy \(\mathrm{S}(\mathrm{aq})\). For a series of solutions having different molalities, comparison of \(\mathrm{E}_{\mathrm{S}}(\mathrm{aq})\) is not straightforward because entropy \(\mathrm{S}(\mathrm{aq})\) is itself a function of solution composition. Further comparison cannot be readily drawn with the isentropic thermal expansion of the pure solvent \(\mathrm{E}_{\mathrm{Sl}}^{*}(\ell)\) equation (e).
\[\mathrm{E}_{\mathrm{S} 1}^{*}(\ell ; \mathrm{A}=0)=\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{A}=0} \text { at constant } \mathrm{S}_{1}^{*}(\ell) \nonumber \]
\(\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0 ; \mathrm{aq})\) is a non-Gibbsian property [2]. Consequently, familiar thermodynamic relationships involving partial molar properties are not valid in the case of partial molar isentropic (thermal) expansions which are non-Lewisian properties [2,3]. \(\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{T}\right]\) for solute-\(j\) in aqueous solution at constant \(\mathrm{S}(\mathrm{aq})\) is a semi-partial molar property [4].
For an aqueous solution having entropy \(\mathrm{S}(\mathrm{aq})\), two partial molar isentropic expansions are defined for the solvent and solute. At \(\mathrm{S}(\mathrm{aq})\) characterised by \(\mathrm{T}\), \(\mathrm{p}\), \(\mathrm{n}_{1}\) and \(\mathrm{n}_{j}\),
\[\mathrm{E}_{\mathrm{Sl}}(\mathrm{aq} ; \mathrm{def})=\left[\partial \mathrm{E}_{\mathrm{S}}(\mathrm{aq}) / \partial \mathrm{n}_{1}\right] \text { at fixed } \mathrm{T}, \mathrm{p} and \mathrm{n}_{j} \nonumber \]
and
\[\mathrm{E}_{\mathrm{Sj}}(\mathrm{aq} ; \mathrm{def})=\left[\partial \mathrm{E}_{\mathrm{S}}(\mathrm{aq}) / \partial \mathrm{n}_{\mathrm{j}}\right] \text { at fixed } \mathrm{T}, \mathrm{p} \text { and } \mathrm{n}_{1} \nonumber \]
So that,
\[\mathrm{E}_{\mathrm{S}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{Sl}}(\mathrm{aq} ; \operatorname{def})+\mathrm{n}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{Sj}}(\mathrm{aq} ; \text { def }) \nonumber \]
Equation (h) relates \(\mathrm{E}_{\mathrm{S}}(\mathrm{aq})\) to the partial molar intensive isentropic properties of both solvent and solute.
A similar problem is encountered in defining an apparent molar isentropic expansion for solute-\(j\), \(\phi\left(\mathrm{E}_{\mathrm{Sj} \mathrm{j}}\right)\). We might assert that \(\phi\left(\mathrm{E}_{\mathrm{Sj} \mathrm{j}}\right)\) is defined by the isentropic differential dependence \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) on temperature. Alternatively, we use an equation by analogy to those used to relate, for example, \(\mathrm{V}(\mathrm{aq})\) to \(V_{1}^{*}(\ell)\) and \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\). Equation (i) relates \(\mathrm{V}(\mathrm{aq})\) to the apparent molar volume of solute j, φ(Vj).
\[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
Differentiation of equation (i) at constant entropy again raises the problem that the molar entropy \(\mathrm{S}(\mathrm{aq})\) does not equal the molar entropy of the pure solvent, \(\mathrm{S}_{1}^{*}(\ell)\). However, by analogy with the definition of \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)\) we define a quantity \(\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \text { def }\right)\) using equation (j).
\[\mathrm{E}_{\mathrm{s}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{Sl}}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right) \nonumber \]
\(\mathrm{E}_{\mathrm{Sl}}^{*}(\ell)\) is the molar intensive property of the solvent. The isentropic expansion of the solution at entropy \(\mathrm{S}(\mathrm{aq})\) is linked with that of the pure solvent at entropy \(\mathrm{S}_{1}^{*}(\ell)\). Further [5]
\[\phi\left(E_{\mathrm{s} j} ; \operatorname{def}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
\[\phi\left(E_{\mathrm{sj}} ; \text { def }\right)=\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right] \,\left(\mathrm{c}_{\mathrm{j}}\right)^{-1}+\alpha_{\mathrm{s} 1}^{*}(1) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
Footnotes
[1] From \(\left(\frac{\partial^{2} \mathrm{U}}{\partial \mathrm{S} \, \partial \mathrm{V}}\right)=\left(\frac{\partial^{2} \mathrm{U}}{\partial \mathrm{V} \, \partial \mathrm{S}}\right), \quad\left(\frac{\partial \mathrm{T}}{\partial \mathrm{V}}\right)_{\mathrm{s}}=-\left(\frac{\partial \mathrm{p}}{\partial \mathrm{S}}\right)_{\mathrm{V}}\) We invert the latter equation. Hence, \(E_{S}=\left(\frac{\partial V}{\partial T}\right)_{s}=-\left(\frac{\partial S}{\partial p}\right)_{V}\)
The isentropic dependence of volume on temperature equals (with reversed sign) the isochoric dependence of entropy on pressure.
[2] J. C. R. Reis, M. J. Blandamer, M. I. Davis and G. Douheret., Phys. Chem. Chem. Phys.,2001, 3 ,1465.
[3] J. C. R. Reis, J. Chem. Soc. Faraday Trans. 2,1982, 78 ,1595.
[4] M. J. Blandamer, M. I. Davis, G. Douheret and J. C. R. Reis., Chem. Soc. Rev., 2001, 30 ,8.
[5] From
\[\begin{aligned}
&\phi\left(E_{\mathrm{S} j} ; \text { def }\right)=\frac{E_{\mathrm{S}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{S} 1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}}} \\
&\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \text { def }\right)=\frac{\mathrm{E}_{\mathrm{S}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)}{\mathrm{m}_{\mathrm{j}}}-\frac{\mathrm{E}_{\mathrm{S} 1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}} \\
&\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)=\frac{\alpha_{\mathrm{s}}(\mathrm{aq}) \, \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)}{\mathrm{m}_{\mathrm{j}}}-\frac{\alpha_{\mathrm{S} 1}^{*}(\ell) \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}}
\end{aligned} \nonumber \]
But \(\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1} / \mathrm{kg}=1\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\)
Then \(\phi\left(\mathrm{E}_{\mathrm{s} j} ; \mathrm{def}\right)=\frac{\alpha_{\mathrm{s}}(\mathrm{aq}) \,\left[\left(1 / \mathrm{M}_{1}\right) \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]}{\mathrm{m}_{\mathrm{j}}}-\frac{\alpha_{\mathrm{s} 1}^{*}(\ell) \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}}\)
Or, \(\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \text { def }\right)=\frac{\mathrm{V}_{1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{sl}}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\)
Hence, \(\phi\left(\mathrm{E}_{\mathrm{s} \mathrm{j}} ; \operatorname{def}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\)
But \(\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}=\frac{1}{\mathrm{c}_{\mathrm{j}}}-\phi\left(\mathrm{V}_{\mathrm{j}}\right)\)
Then \phi\left(E_{s j} ; \operatorname{def}\right)=\left[\frac{1}{c_{j}}-\phi\left(V_{j}\right)\right] \,\left[\alpha_{s}(\mathrm{aq})-\alpha_{\mathrm{s}_{1}}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\)
Or, \(\begin{aligned}
\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right] \\
&-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s}}(\mathrm{aq})+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s} 1}^{*}(\ell)+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned}\)
Hence,
\[\begin{aligned}
&\phi\left(\mathrm{E}_{\mathrm{sj}} ; \mathrm{def}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s} 1}^{*}(\ell) \\
\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \text { def }\right)=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]
\end{aligned} \nonumber \]